Many-body localization in a slowly varying potential

We study many-body localization (MBL) in a nearest-neighbor hopping 1D lattice with a slowly varying (SV) on-site potential $U_j = \lambda\cos(\pi\alpha j^s)$ with $0<s<1$. The corresponding non-interacting 1D lattice model is known to have single-particle localization with mobility edges. Using exact diagonalization, we find that the MBL of this model has similar features to the conventional MBL of extensively studied random or quasiperiodic (QP) models, including the transitions of eigenstate entanglement entropy (EE) and level statistics, and the logarithmic growth of EE. To further investigate the universal properties of this MBL transition in the asymptotic regime, we implement a real-space renormalization group (RG) method. RG analysis shows a subvolume scaling $\sim L^{d_{\rm MBL}}$ with $d_{\rm MBL} \approx 1-s$ of the localization length (length of the largest thermal clusters) in this MBL phase. In addition, we explore the critical properties and find universal scalings of the EE and localization length. From these quantities, we compute the critical exponent $\nu$ using different parameters $s$ (characterizing different degrees of spatial variation of the imposed potential), finding the critical exponent staying around $\nu\approx2$. This exponent $\nu \approx 2$ is close to that of the QP model within the error bars but differs from the random model. This observation suggests that the SV and QP models may belong to the same universality class, which is, however, likely distinct from the random universality class.